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In his seminal paper on Special Relativity: On the Electrodynamics of Moving Bodies, Einstein derives a formula for the Relativistic Doppler effect.
See this section:
§ 7. Theory of Doppler's Principle and of AberrationThe formula is:
In this Wiki article, the same formula is derived.
The article uses the "the standard Lorentz transformation of the four-momentum".
Now here's where it gets a little muddling.
Einstein writes:
It seems that his formula is a good approximation when we are sufficiently far from the source.
However, the four-momentum transformations do not depend on distance, yet they arrive at the exact same formula.
What's the reason for this difference?
See this section:
§ 7. Theory of Doppler's Principle and of AberrationThe formula is:
The article uses the "the standard Lorentz transformation of the four-momentum".
Now here's where it gets a little muddling.
Einstein writes:
Albert Einstein said:In the system K, very far from the origin of co-ordinates, let there be a source of electrodynamic waves, which in a part of space containing the origin of co-ordinates may be represented to a sufficient degree of approximation by the equations
Albert Einstein said:...if an observer is moving with velocity v relatively to an infinitely distant source of light...
It seems that his formula is a good approximation when we are sufficiently far from the source.
However, the four-momentum transformations do not depend on distance, yet they arrive at the exact same formula.
What's the reason for this difference?